An axiomatic framework from splitting and merging in MAT-labeled graphs, vines, and single-peaked domains

Hung Manh Tran; Shuhei Tsujie; Tan Nhat Tran

arxiv: 2605.20629 · v1 · pith:D5ZLJJU2new · submitted 2026-05-20 · 🧮 math.CO

An axiomatic framework from splitting and merging in MAT-labeled graphs, vines, and single-peaked domains

Hung Manh Tran , Tan Nhat Tran , Shuhei Tsujie This is my paper

Pith reviewed 2026-05-21 04:23 UTC · model grok-4.3

classification 🧮 math.CO

keywords MAT-labeled graphsregular vinessingle-peaked domainssplitting and mergingcombinatorial speciesaxiomatic characterizationsocial choice theoryformal concept analysis

0 comments

The pith

MAT-labeled complete graphs, regular vines, and maximal single-peaked domains arise from the same recursive combinatorial structure via splitting and merging operations with compatibility conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that MAT-labeled complete graphs from hyperplane arrangements, regular vines from probability theory, and maximal Arrow single-peaked domains from social choice theory all emerge from one recursive combinatorial structure. This structure is defined by splitting and merging operations together with natural compatibility conditions that uniquely determine each family of objects. A sympathetic reader would care because the shared axioms deliver explicit correspondences among the three structures and supply the combinatorial characterization of single-peaked domains that had remained open in economics. The same framework also accommodates (n,3)-extremal lattices from formal concept analysis, showing the pattern is not isolated to one field.

Core claim

MAT-labeled complete graphs, regular vines, and maximal Arrow's single-peaked domains arise from the same recursive combinatorial structure defined by splitting and merging operations together with natural compatibility conditions that uniquely determine the structures. The main result gives an axiomatic characterization of these objects using the language of combinatorial species.

What carries the argument

splitting and merging operations with natural compatibility conditions inside the framework of combinatorial species

If this is right

Explicit correspondences exist between maximal Arrow's single-peaked domains, MAT-labeled complete graphs, and regular vines.
Regular vines are equivalent to (n,3)-extremal lattices from formal concept analysis.
The extremal lattices also satisfy the splitting and merging axiomatic characterization.
The open problem on combinatorial characterization of single-peaked domains in economics is resolved.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Methods developed for counting or sampling regular vines could now be applied directly to enumerate maximal single-peaked domains.
The recursive description may yield polynomial-time recognition algorithms for these objects in each of the three settings.
Analogous splitting and merging axioms could be tested on other families of combinatorial objects that appear in social choice or lattice theory.

Load-bearing premise

The compatibility conditions on the splitting and merging operations are sufficient to uniquely characterize the maximal single-peaked domains as they are standardly defined in social choice theory.

What would settle it

A maximal single-peaked domain that cannot be constructed by any finite sequence of the splitting and merging operations, or a structure produced by those operations that fails to be single-peaked under the usual definition.

read the original abstract

In recent work (Forum Math.~Sigma, 2024), we established a correspondence between MAT-labeled graphs arising from hyperplane arrangements and regular vines from probability theory. In this paper, we extend this connection to Arrow's single-peaked domains in social choice theory. We show that MAT-labeled complete graphs, regular vines, and maximal Arrow's single-peaked domains arise from the same recursive combinatorial structure. Our main result gives an axiomatic characterization of these objects using the language of combinatorial species. At the heart of this characterization are two fundamental operations, called splitting and merging, together with natural compatibility conditions that uniquely determine the structures. As consequences, we obtain explicit correspondences between maximal Arrow's single-peaked domains, MAT-labeled complete graphs, and regular vines, resolving an open problem in the economics literature concerning the combinatorial characterization of single-peaked domains. We further show, by a direct proof, that regular vines are equivalent to $(n,3)$-extremal lattices from formal concept analysis. Consequently, these extremal lattices also fit naturally into the same splitting and merging framework, providing another example from a different area that satisfies our axiomatic characterization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper unifies MAT-labeled graphs, vines, and single-peaked domains via splitting and merging axioms, with a claimed resolution of an open combinatorial problem in economics.

read the letter

The main takeaway is that the authors extend their 2024 Forum Math. Sigma result by showing MAT-labeled complete graphs, regular vines, and maximal Arrow single-peaked domains all arise from the same recursive splitting and merging operations under natural compatibility conditions. They give an axiomatic characterization in the language of combinatorial species and claim this uniquely pins down the structures. As a side result they prove regular vines are equivalent to (n,3)-extremal lattices by direct argument, folding that class into the same framework too. This supplies explicit correspondences across the areas and addresses the open question on combinatorial descriptions of single-peaked domains. The unification itself is the clearest new piece; the lattice equivalence looks like a clean, self-contained addition. The compatibility conditions and uniqueness claims are plausible from the setup, and the paper treats the prior MAT-vine correspondence as a base rather than re-deriving it circularly. The soft spot sits in the verification that the axiomatic class matches the classical maximal single-peaked domains exactly. The abstract indicates they establish this by showing the generated objects satisfy the single-peaked property and maximality, but the details of the argument—whether by induction, forbidden subconfiguration check, or explicit bijection—matter for whether extra objects slip in or some standard domains are missed. A few small-n examples or a direct comparison table would make that step easier to inspect. Readers working at the intersection of combinatorics, vine copulas, or social choice theory will get the most from it; the cross-field links are concrete enough to be usable for further work. The paper is coherent on its own terms and shows honest engagement with the cited literature, so it deserves a serious referee even if the equivalence proof needs some polishing in review.

Referee Report

2 major / 2 minor

Summary. The paper extends a prior correspondence between MAT-labeled graphs and regular vines to maximal Arrow single-peaked domains. It introduces splitting and merging operations on combinatorial species together with compatibility conditions, claims these axioms uniquely determine the three classes of objects, establishes explicit correspondences among them, and proves by direct argument that regular vines coincide with (n,3)-extremal lattices.

Significance. If the central equivalences are fully established, the work supplies a unified recursive description that links hyperplane-arrangement combinatorics, vine copula models, social-choice theory, and formal concept analysis while resolving an open combinatorial characterization question for maximal single-peaked domains. The direct proof relating vines to extremal lattices is a concrete strength.

major comments (2)

[Main Theorem / Section 4] Main result (the axiomatic characterization theorem): the manuscript must supply an explicit argument—whether by induction on n, by exhibiting a bijection, or by verifying the single-peaked property plus maximality—that the objects generated by the splitting/merging operations with the stated compatibility conditions coincide exactly with the classical maximal single-peaked domains (i.e., those avoiding the standard forbidden subconfigurations). The abstract and result statement leave this direction implicit.
[Section 5] §5 (or the section containing the vine–lattice equivalence): while a direct proof is announced, the argument should be checked for any hidden appeal to the authors’ 2024 MAT-vine correspondence; if the new axiomatization is intended to be independent, the proof must not presuppose that earlier result.

minor comments (2)

[Abstract] Abstract: the phrase 'resolving an open problem in the economics literature' would benefit from a one-sentence reminder of the precise open question being settled.
[Section 3] Notation: the compatibility conditions are described as 'natural'; a short table or enumerated list making each condition explicit would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments identify places where the exposition of the main results can be strengthened, and we will make the indicated revisions to improve clarity and completeness.

read point-by-point responses

Referee: [Main Theorem / Section 4] Main result (the axiomatic characterization theorem): the manuscript must supply an explicit argument—whether by induction on n, by exhibiting a bijection, or by verifying the single-peaked property plus maximality—that the objects generated by the splitting/merging operations with the stated compatibility conditions coincide exactly with the classical maximal single-peaked domains (i.e., those avoiding the standard forbidden subconfigurations). The abstract and result statement leave this direction implicit.

Authors: We agree that the direction establishing exact coincidence with classical maximal Arrow single-peaked domains (via avoidance of forbidden configurations) is currently stated as a consequence rather than proved in full detail. In the revised manuscript we will add an explicit inductive argument in Section 4. The proof proceeds by induction on n: the base cases are verified directly, and the inductive step shows that splitting and merging preserve both the single-peaked property and maximality while generating all objects that avoid the standard forbidden subconfigurations. This will render the axiomatic characterization fully rigorous. revision: yes
Referee: [Section 5] §5 (or the section containing the vine–lattice equivalence): while a direct proof is announced, the argument should be checked for any hidden appeal to the authors’ 2024 MAT-vine correspondence; if the new axiomatization is intended to be independent, the proof must not presuppose that earlier result.

Authors: The direct proof in Section 5 relating regular vines to (n,3)-extremal lattices is self-contained and makes no reference to the 2024 MAT-vine correspondence. It relies only on the splitting/merging operations and lattice-theoretic properties introduced in the present paper. To make this independence explicit for readers, we will add a short clarifying remark at the start of the section in the revised version. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to 2024 MAT-vine result; new axiomatic splitting/merging framework is independent

full rationale

The paper cites its own 2024 Forum Math. Sigma work to recall the MAT-labeled graph / regular vine correspondence, then introduces splitting and merging operations plus compatibility conditions as a new combinatorial species axiomatization. The main result claims these conditions uniquely determine the structures and thereby equate them to maximal single-peaked domains, with the equivalence presented as a direct consequence rather than a redefinition or fit of prior parameters. No equation reduces a claimed prediction to a fitted input by construction, and the uniqueness statement is not imported solely from the self-citation; the derivation chain for the single-peaked characterization therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities are described. The central claim rests on the prior MAT-vine correspondence (self-cited) and on the domain assumption that the stated compatibility conditions suffice for uniqueness.

axioms (1)

domain assumption Compatibility conditions on splitting and merging operations uniquely determine the combinatorial structures of interest.
This uniqueness is the load-bearing step of the axiomatic characterization described in the abstract.

pith-pipeline@v0.9.0 · 5743 in / 1356 out tokens · 67790 ms · 2026-05-21T04:23:02.377686+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Our main result gives an axiomatic characterization of these objects using the language of combinatorial species. At the heart of this characterization are two fundamental operations, called splitting and merging, together with natural compatibility conditions that uniquely determine the structures. (Theorem 3.7)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean equivNat / recovery theorem echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Theorem 3.7. ... the pair (F, σ_F) is unique ... η is a natural isomorphism.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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T. Abe, M. Barakat, M. Cuntz, T. Hoge, and H. Terao. The freeness of ideal subarrangements of Weyl arrange- ments.J. Eur. Math. Soc. (JEMS), 18(6):1339–1348, 2016

work page 2016
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A. Albano and B. Chornomaz. Why concept lattices are large: Extremal theory for generators, concepts, and VC-dimension.Int. J. Gen. Syst., 46(5):440–457, 2017

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T. Bedford and R. M. Cooke. Vines—a new graphical model for dependent random variables.Ann. Statist., 30(4):1031–1068, 2002

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F. Bergeron, G. Labelle, and P. Leroux.Combinatorial species and tree-like structures, volume 67 ofEncyclo- pedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1998. Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota

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B. Chornomaz. Counting extremal lattices. HAL preprint, 2016.https://hal.science/ hal-01175633v2

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R. M. Cooke, D. Kurowicka, and K. Wilson. Sampling, conditionalizing, counting, merging, searching regular vines.J. Multivariate Anal., 138:4–18, 2015

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K. Inada. A note on the simple majority decision rule.Econometrica, 32:525–531, 1964

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H. Joe, R. M. Cooke, and D. Kurowicka. Regular Vines: Generation Algorithm and Number of Equivalence Classes. In D. Kurowicka and H. Joe, editors,Dependence Modeling, pages 219–231. WORLD SCIENTIFIC, Dec. 2010

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Karpov and A

A. Karpov and A. Slinko. Constructing large peak-pit Condorcet domains.Theory and Decision, 94(1):97–120, 2023

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Kurowicka and R

D. Kurowicka and R. M. Cooke. Completion problem with partial correlation vines.Linear Algebra Appl., 418(1):188–200, 2006. 32 HUNG MANH TRAN, TAN NHAT TRAN, AND SHUHEI TSUJIE

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Kurowicka and H

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Markstr ¨om, S

K. Markstr ¨om, S. Riis, and B. Zhou. Arrow’s single peaked domains, richness, and domains for plurality and the Borda count. arXiv preprint, 2024.https://arxiv.org/abs/2401.12547

work page arXiv 2024
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Monjardet

B. Monjardet. Acyclic domains of linear orders: A survey. InThe mathematics of preference, choice and order, Stud. Choice Welf., pages 139–160. Springer, Berlin, 2009

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Morales-N ´apoles

O. Morales-N ´apoles. Counting Vines. In D. Kurowicka and H. Joe, editors,Dependence Modeling, pages 189–

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WORLD SCIENTIFIC, Dec. 2010

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[21]

Orlik and H

P. Orlik and H. Terao.Arrangements of hyperplanes, volume 300 ofGrundlehren der mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992

work page 1992
[22]

C. Puppe. The single-peaked domain revisited: A simple global characterization.J. Econom. Theory, 176:55–80, 2018

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A. Slinko. Condorcet domains satisfying Arrow’s single-peakedness.J. Math. Econom., 84:166–175, 2019

work page 2019
[24]

A. Slinko. A combinatorial representation of Arrow’s single-peaked domains. arXiv preprint, 2024.https: //arxiv.org/abs/2412.05406

work page arXiv 2024
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N. J. A. Sloane and T. O. F. Inc. The on-line encyclopedia of integer sequences.https://oeis.org/, 2010

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Sommers and J

E. Sommers and J. Tymoczko. Exponents forB-stable ideals.Trans. Amer. Math. Soc., 358(8):3493–3509, 2006

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[27]

H. Terao. Arrangements of hyperplanes and their freeness. I.J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27(2):293– 312, 1980

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[28]

H. M. Tran, T. N. Tran, and S. Tsujie. Vines and MAT-labeled graphs.Forum Math. Sigma, 12:Paper No. e128, 29, 2024

work page 2024
[29]

T. N. Tran and S. Tsujie. MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling.Algebr. Comb., 6(6):1447–1467, 2023

work page 2023
[30]

Zhu and D

K. Zhu and D. Kurowicka. Regular vines with strongly chordal pattern of (conditional) independence.Comput. Statist. Data Anal., 172:Paper No. 107461, 24, 2022. HUNGMANHTRAN, FACULTY OFFUNDAMENTALSCIENCES, PHENIKAAUNIVERSITY, HANOI12116, VIET- NAM. Email address:hung.tranmanh@phenikaa-uni.edu.vn TANNHATTRAN, DEPARTMENT OFMATHEMATICS ANDSTATISTICS, BINGHAMT...

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[1] [1]

T. Abe, M. Barakat, M. Cuntz, T. Hoge, and H. Terao. The freeness of ideal subarrangements of Weyl arrange- ments.J. Eur. Math. Soc. (JEMS), 18(6):1339–1348, 2016

work page 2016

[2] [2]

Albano and B

A. Albano and B. Chornomaz. Why concept lattices are large: Extremal theory for generators, concepts, and VC-dimension.Int. J. Gen. Syst., 46(5):440–457, 2017

work page 2017

[3] [3]

K. J. Arrow.Social Choice and Individual Values. second edition John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1963

work page 1963

[4] [4]

Bedford and R

T. Bedford and R. M. Cooke. Vines—a new graphical model for dependent random variables.Ann. Statist., 30(4):1031–1068, 2002

work page 2002

[5] [5]

Bergeron, G

F. Bergeron, G. Labelle, and P. Leroux.Combinatorial species and tree-like structures, volume 67 ofEncyclo- pedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1998. Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota

work page 1998

[6] [6]

D. Black. On the rationale of group decision-making.J. Political Economy, 56(1):23–34, 1948

work page 1948

[7] [7]

Chornomaz

B. Chornomaz. Counting extremal lattices. HAL preprint, 2016.https://hal.science/ hal-01175633v2

work page 2016

[8] [8]

R. M. Cooke, D. Kurowicka, and K. Wilson. Sampling, conditionalizing, counting, merging, searching regular vines.J. Multivariate Anal., 138:4–18, 2015

work page 2015

[9] [9]

Cuntz and P

M. Cuntz and P. M ¨ucksch. MAT-free reflection arrangements.Electron. J. Combin., 27(1):Paper No. 1.28, 19, 2020

work page 2020

[10] [10]

P. H. Edelman and V . Reiner. Free hyperplane arrangements betweenA n−1 andB n.Math. Z., 215(3):347–365, 1994

work page 1994

[11] [11]

K. Inada. A note on the simple majority decision rule.Econometrica, 32:525–531, 1964

work page 1964

[12] [12]

H. Joe, R. M. Cooke, and D. Kurowicka. Regular Vines: Generation Algorithm and Number of Equivalence Classes. In D. Kurowicka and H. Joe, editors,Dependence Modeling, pages 219–231. WORLD SCIENTIFIC, Dec. 2010

work page 2010

[13] [13]

Karpov and A

A. Karpov and A. Slinko. Constructing large peak-pit Condorcet domains.Theory and Decision, 94(1):97–120, 2023

work page 2023

[14] [14]

Kurowicka and R

D. Kurowicka and R. M. Cooke. Completion problem with partial correlation vines.Linear Algebra Appl., 418(1):188–200, 2006. 32 HUNG MANH TRAN, TAN NHAT TRAN, AND SHUHEI TSUJIE

work page 2006

[15] [15]

Kurowicka and H

D. Kurowicka and H. Joe, editors.Dependence modeling. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. Vine copula handbook

work page 2011

[16] [16]

Liversidge

G. Liversidge. Counting condorcet domains. arXiv preprint, 2020.https://arxiv.org/abs/2004. 00751

work page 2020

[17] [17]

Markstr ¨om, S

K. Markstr ¨om, S. Riis, and B. Zhou. Arrow’s single peaked domains, richness, and domains for plurality and the Borda count. arXiv preprint, 2024.https://arxiv.org/abs/2401.12547

work page arXiv 2024

[18] [18]

Monjardet

B. Monjardet. Acyclic domains of linear orders: A survey. InThe mathematics of preference, choice and order, Stud. Choice Welf., pages 139–160. Springer, Berlin, 2009

work page 2009

[19] [19]

Morales-N ´apoles

O. Morales-N ´apoles. Counting Vines. In D. Kurowicka and H. Joe, editors,Dependence Modeling, pages 189–

work page

[20] [20]

WORLD SCIENTIFIC, Dec. 2010

work page 2010

[21] [21]

Orlik and H

P. Orlik and H. Terao.Arrangements of hyperplanes, volume 300 ofGrundlehren der mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992

work page 1992

[22] [22]

C. Puppe. The single-peaked domain revisited: A simple global characterization.J. Econom. Theory, 176:55–80, 2018

work page 2018

[23] [23]

A. Slinko. Condorcet domains satisfying Arrow’s single-peakedness.J. Math. Econom., 84:166–175, 2019

work page 2019

[24] [24]

A. Slinko. A combinatorial representation of Arrow’s single-peaked domains. arXiv preprint, 2024.https: //arxiv.org/abs/2412.05406

work page arXiv 2024

[25] [25]

N. J. A. Sloane and T. O. F. Inc. The on-line encyclopedia of integer sequences.https://oeis.org/, 2010

work page 2010

[26] [26]

Sommers and J

E. Sommers and J. Tymoczko. Exponents forB-stable ideals.Trans. Amer. Math. Soc., 358(8):3493–3509, 2006

work page 2006

[27] [27]

H. Terao. Arrangements of hyperplanes and their freeness. I.J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27(2):293– 312, 1980

work page 1980

[28] [28]

H. M. Tran, T. N. Tran, and S. Tsujie. Vines and MAT-labeled graphs.Forum Math. Sigma, 12:Paper No. e128, 29, 2024

work page 2024

[29] [29]

T. N. Tran and S. Tsujie. MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling.Algebr. Comb., 6(6):1447–1467, 2023

work page 2023

[30] [30]

Zhu and D

K. Zhu and D. Kurowicka. Regular vines with strongly chordal pattern of (conditional) independence.Comput. Statist. Data Anal., 172:Paper No. 107461, 24, 2022. HUNGMANHTRAN, FACULTY OFFUNDAMENTALSCIENCES, PHENIKAAUNIVERSITY, HANOI12116, VIET- NAM. Email address:hung.tranmanh@phenikaa-uni.edu.vn TANNHATTRAN, DEPARTMENT OFMATHEMATICS ANDSTATISTICS, BINGHAMT...

work page 2022